Morphological Image Processing (2)

Topics Morphological Operations

Connected Component Extraction

Convex Hull

Thinning

Thickening

Skeleton

Pruning

Extension to gray level images

Dilation and Erosion Dilation and Erosion are two basic operations in

morphological processing.

Dilation of a set A in Z2 by a set B in Z2 is denoted by A B and given by

Dilation The dilation of A by B is the set of all displacements

such that A and overlap with at least one point

B is called structuring element

Dilation

Erosion Erosion of a set A in Z2 by a set B in Z2 is denoted by

and given by:

Erosion of A be B is the set of all points z such that B translated by z is contained in A

Erosion

Opening Opening smoothes the outer contours, breaks narrow

connections, and eliminates small protrusions.

Opening is defined as :

Closing Closing smoothes the object contour, fuses narrow

connections, eliminates small holes and gaps.

Extraction of Connected Components Begin with a point P inside the connected component,

iterate:

Until Xk = Xk-1

Initially X0 = P

Connected Component Extraction

Convex Hull A set is said convex if the straight line connecting any

two points of the set lies entirely within A.

Convex Hull of set S is the smallest convex set A that contains S

The set difference A-S is called the convex deficiency of S

Computing Convex Hull Let Bi for i=1,2,3,4 represent the structuring elements

shown below

Convex Hull Repeat the following equation until converge

with

is the Hit-or-Miss operator

Assuming Convex Hull is 4

1

)(

i

iDAC

Example (Convex Hull)

Improving Convex Hull Algorithm The algorithm can be improved by limiting the growth

of the algorithm beyond the maximum dimensions of the original set.

Thinning and Thickening Thinning is an image-processing operation in which

binary valued image regions are reduced to lines

The purpose of thinning is to reduce the image components to their essential information for further analysis and recognition

Thickening is changing a pixel from 1 to 0 if any neighbors of the pixel are 1.

Thickening followed by thinning can be used for filling undesirable holes.

Thinning followed by thickening is used for determining isolated components and clusters.

Thinning Thinning is defined in terms of hit or miss as

where B is a sequence of structuring elements like

{B} = {B1, B2, B3, …, Bn} and the operation can be given as

Thinning Sample set of structuring elements

Thinning Example

Thickening Thickening is the morphological dual of thinning and

defined as

or

Thickening Example

Skeleton The informal definition of a skeleton is a line

representation of an object that is:

one-pixel thick,

through the "middle" of the object, and,

preserves the topology of the object.

Skeleton Skeleton is defined by

where

k is the last iterative step before A erodes to an empty set

Skeleton Example

Pruning Thinning and skeletonizing algorithms need a clean-up

post-processing

The following steps are used for pruning:

Thinning

Restore to original form with end points removed

Dilate end points

Find the union of X1 and X3

Pruning Example Original image and structuring elements

Pruning Example Result of thinning and end points detected

Pruning Example Dilation of end points and the pruned image

Extension to Gray Level Dilation is expressed in 1D as

Erosion is given by

Extension to Gray Level (2D Case) Dilation

Erosion

Questions?